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c_s4z6h24peci9 | In nuclear warfare theory, a decapitation strike is a pre-emptive first strike attack that aims to destabilize an opponent's military and civil leadership structure in the hope that it will severely degrade or destroy its capacity for nuclear retaliation. It is essentially a subset of a counterforce strike but whereas ... | Decapitation strike |
c_ni8vjpb9pnrg | Delegation of ICBM/SLBM launch capability to local commanders in the event of a decapitation strike. Distributed and diverse launch mechanisms.A failed decapitation strike carries the risk of immediate, massive retaliation by the targeted opponent. Many countries with nuclear weapons specifically plan to prevent decapi... | Decapitation strike |
c_inwqbmiksmq2 | Such countries may have mobile land-based launch, sea launch, air launch, and underground ballistic missile launch facilities so that a nuclear attack on one area of the country will not totally negate its ability to retaliate. Other nuclear warfare doctrines explicitly exclude decapitation strikes on the basis that it... | Decapitation strike |
c_pk6855l9ztbq | In nuclear warfare, enemy targets are divided into two types: counterforce and countervalue. A counterforce target is an element of the military infrastructure, usually either specific weapons or the bases that support them. A counterforce strike is an attack that targets those elements but leaving the civilian infrast... | Counterforce strike |
c_kccmnck6ltud | An ideal counterforce attack would kill no civilians. Military attacks are prone to causing collateral damage, especially when nuclear weapons are employed. | Counterforce strike |
c_x4iarvo88bu1 | In nuclear terms, many military targets are located near civilian centers, and a major counterforce strike that uses even relatively small nuclear warheads against a nation would certainly inflict many civilian casualties. Also, the requirement to use ground burst strikes to destroy hardened targets would produce far m... | Counterforce strike |
c_h7a1a5cxxo18 | In nuclear weapon design, the pit is the core of an implosion nuclear weapon, consisting of fissile material and any neutron reflector or tamper bonded to it. Some weapons tested during the 1950s used pits made with uranium-235 alone, or as a composite with plutonium. All-plutonium pits are the smallest in diameter and... | Plutonium core |
c_eq5ev0cgwm1j | In nuclei, the entire assembly of protons and neutrons (nucleons) has a resultant angular momentum due to the angular momenta of each nucleon, usually denoted I. If the total angular momentum of a neutron is jn = ℓ + s and for a proton is jp = ℓ + s (where s for protons and neutrons happens to be 1/2 again (see note)),... | Quantum numbers |
c_xpehcs347o2n | In nucleic acids nanotechnology, artificial nucleic acids are designed to form molecular components that can self-assemble into stable structures for use ranging from targeted drug delivery to programmable biomaterials. DNA nanotechnology uses DNA motifs to build target shapes and arrangements. It has been used in a va... | RNA origami |
c_94hbwtxxerag | RNA nanotechnology combines the simplistic design and manipulation characteristic of DNA, with the additional flexibility in structure and diversity in function similar to that of proteins. RNA's versatility in structure and function, favorable in vivo attributes, and bottom-up self-assembly is an ideal avenue for deve... | RNA origami |
c_w24wn2j6dg3q | The first work in RNA origami appeared in Science, published by Ebbe S. Andersen of Aarhus University. Researchers at Aarhus University used various 3D models and computer software to design individual RNA origami. | RNA origami |
c_p7nv9qsecxff | Once encoded as a synthetic DNA gene, adding RNA polymerase resulted in the formation of RNA origami. Observation of RNA was primarily done through atomic force microscopy, a technique that allows researchers to look at molecules a thousand times closer than would normally be possible with a conventional light microsco... | RNA origami |
c_s9fp66rovciv | They were able to form honeycomb shapes, but determined other shapes are also possible. Cody Geary, a scholar in the field of RNA origami, described the uniqueness of the method of RNA origami. He stated that its folding recipe is encoded in the molecule itself, and determine by its sequence. The sequence gives the RNA... | RNA origami |
c_7vei0n8mrldr | In nucleophilic aliphatic substitution, sodium nitrite (NaNO2) replaces an alkyl halide. In the so-called Ter Meer reaction (1876) named after Edmund ter Meer, the reactant is a 1,1-halonitroalkane: The reaction mechanism is proposed in which in the first slow step a proton is abstracted from nitroalkane 1 to a carbani... | Nitro compounds |
c_sxggpsxej36q | In nucleophilic trifluoromethylation the active species is the CF3− anion. It was, however, widely believed that the trifluoromethyl anion is a transient species and thus cannot be isolated or observed in the condensed phase. Contrary to the popular belief, the CF3 anion, with + as a countercation, was produced and cha... | Trifluoromethylation |
c_9w1wm6uksmnd | The challenges associated with observation of CF3 anion are alluded to its strong basic nature and its tendency to form pentacoordinated silicon species, such as − or −. The reactivity of fluoroform in combination with a strong base such as t-BuOK with carbonyl compounds in DMF is an example. Here CF3− and DMF form an ... | Trifluoromethylation |
c_z1gx99sow9vi | In nucleotide sugar metabolism a group of biochemicals known as nucleotide sugars act as donors for sugar residues in the glycosylation reactions that produce polysaccharides. They are substrates for glycosyltransferases. The nucleotide sugars are also intermediates in nucleotide sugar interconversions that produce som... | Nucleotide sugars metabolism |
c_fpurlxnt4d1p | These glycoconjugates are virulence factors and components of the fungal and bacterial cell wall. These pathways are also studied in plants, but here the enzymes involved are less well understood. == References == | Nucleotide sugars metabolism |
c_g6f09uq4uyat | In null-hypothesis significance testing, the p-value is the probability of obtaining test results at least as extreme as the result actually observed, under the assumption that the null hypothesis is correct. A very small p-value means that such an extreme observed outcome would be very unlikely under the null hypothes... | P value |
c_fx7j52owf8wy | In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings. An ideal in the ring of integers of an algebraic number field is principal if it consists of... | Ideal number |
c_1mt2rplluvxw | In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used t... | Modular curves |
c_fwk9flxwx5tw | In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. ... | Rational point |
c_7agri53m5njn | In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The conjecture is a central problem in the theory of algebraic cycles. It can be co... | Tate conjecture |
c_5px6jkiiu7sg | In number theory and algebraic geometry, the Tate twist, named after John Tate, is an operation on Galois modules. For example, if K is a field, GK is its absolute Galois group, and ρ: GK → AutQp(V) is a representation of GK on a finite-dimensional vector space V over the field Qp of p-adic numbers, then the Tate twist... | Tate twist |
c_tdbuc7y3wokh | Denoting by Qp(−1) the dual representation of Qp(1), the -mth Tate twist of V can be defined as V ⊗ Q p ( − 1 ) ⊗ m . {\displaystyle V\otimes \mathbf {Q} _{p}(-1)^{\otimes m}.} == References == | Tate twist |
c_93a7dib3gi0c | In number theory and combinatorics rank of a partition of a positive integer is a certain integer associated with the partition. Dyson introduced the concept in a paper published in the journal Eureka. It was presented in the context of a study of certain congruence properties of the partition function discovered by th... | Freeman Dyson |
c_0vb1gv4y5jam | In number theory and combinatorics, a multipartition of a positive integer n is a way of writing n as a sum, each element of which is in turn a partition. The concept is also found in the theory of Lie algebras. | Multipartition |
c_dybck3tnuldi | In number theory and combinatorics, a partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, ... | Ferrers diagram |
c_m96i3trpku8p | The order-dependent composition 1 + 3 is the same partition as 3 + 1, and the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition as 2 + 1 + 1. An individual summand in a partition is called a part. The number of partitions of n is given by the partition function p(n). | Ferrers diagram |
c_pr119f4uf66j | So p(4) = 5. The notation λ ⊢ n means that λ is a partition of n. Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in... | Ferrers diagram |
c_csdc6n5h9f59 | In number theory and computer science, the partition problem, or number partitioning, is the task of deciding whether a given multiset S of positive integers can be partitioned into two subsets S1 and S2 such that the sum of the numbers in S1 equals the sum of the numbers in S2. Although the partition problem is NP-com... | Partition problem |
c_emxv1x7cfw8y | In multiway number partitioning, there is an integer parameter k, and the goal is to decide whether S can be partitioned into k subsets of equal sum (the partition problem is the special case in which k = 2). However, it is quite different than the 3-partition problem: in that problem, the number of subsets is not fixe... | Partition problem |
c_bp14pupkn01r | In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the number of weak orderings on a set of n {\displaystyle n} elements. Weak orderings arrange their elements into a sequence allowing ties, such as might arise as the outcome of a horse race). Starting from n = 0 {\displays... | Ordered Bell number |
c_ax3nm32xl8x5 | In number theory and mathematical logic, a Meertens number in a given number base b {\displaystyle b} is a natural number that is its own Gödel number. It was named after Lambert Meertens by Richard S. Bird as a present during the celebration of his 25 years at the CWI, Amsterdam. | Meertens number |
c_gpbfpx4aqhki | In number theory and set theory, the minimum overlap problem is a problem proposed by Hungarian mathematician Paul Erdős in 1955. | Minimum overlap problem |
c_xp9x4jqivizk | In number theory another arithmetic function closely related to the Möbius function is the Mertens function, defined by M ( n ) = ∑ k = 1 n μ ( k ) {\displaystyle M(n)=\sum _{k=1}^{n}\mu (k)} for every natural number n. This function is closely linked with the positions of zeroes of the Riemann zeta function. See the a... | Moebius function |
c_4slfpab4hxif | In number theory one may study a Diophantine equation, for example, modulo p for all primes p, looking for constraints on solutions. The next step is to look modulo prime powers, and then for solutions in the p-adic field. This kind of local analysis provides conditions for solution that are necessary. In cases where l... | Local analysis |
c_iuil3b0w2y4a | It does for quadratic forms, but certainly not in general (for example for elliptic curves). The point of view that one would like to understand what extra conditions are needed has been very influential, for example for cubic forms. Some form of local analysis underlies both the standard applications of the Hardy–Litt... | Local analysis |
c_vw8h1w52be19 | In number theory the Agoh–Giuga conjecture on the Bernoulli numbers Bk postulates that p is a prime number if and only if p B p − 1 ≡ − 1 ( mod p ) . {\displaystyle pB_{p-1}\equiv -1{\pmod {p}}.} It is named after Takashi Agoh and Giuseppe Giuga. | Agoh–Giuga conjecture |
c_1zoeyb1yfjdg | In number theory the n conjecture is a conjecture stated by Browkin & Brzeziński (1994) as a generalization of the abc conjecture to more than three integers. | N conjecture |
c_zcd9t4gy4391 | In number theory, "almost all positive integers" can mean "the positive integers in a set whose natural density is 1". That is, if A is a set of positive integers, and if the proportion of positive integers in A below n (out of all positive integers below n) tends to 1 as n tends to infinity, then almost all positive i... | Almost all |
c_k45g76bdo7w9 | Almost all even positive numbers can be expressed as the sum of two primes. : 489 Almost all primes are isolated. Moreover, for every positive integer g, almost all primes have prime gaps of more than g both to their left and to their right; that is, there is no other prime between p − g and p + g. | Almost all |
c_xczlav2l3799 | In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, coefficients ai of the continued fraction expansion of x have a finite geometric mean that is independent of the value of x and is known as Khinchin's constant. That is, for x = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1 ⋱ {\displaystyle x... | Khinchin's constant |
c_tpftg41pyh5f | a n ) 1 / n = K 0 {\displaystyle \lim _{n\rightarrow \infty }\left(a_{1}a_{2}...a_{n}\right)^{1/n}=K_{0}} where K 0 {\displaystyle K_{0}} is Khinchin's constant K 0 = ∏ r = 1 ∞ ( 1 + 1 r ( r + 2 ) ) log 2 r ≈ 2.6854520010 … {\displaystyle K_{0}=\prod _{r=1}^{\infty }{\left(1+{1 \over r(r+2)}\right)}^{\log _{2}r}\appr... | Khinchin's constant |
c_l95ohw9byp9v | In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple th... | Artin constant |
c_cavr76z7ck2a | In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over a field Z p {\displaystyle \mathbb {Z} _{p}} . The method was discovered by Elwyn Berlekamp in 1970 as an auxiliary to the algorithm for polynomial factorizat... | Berlekamp–Rabin algorithm |
c_tj8sb8o6po5y | In number theory, Bertrand's postulate is a theorem stating that for any integer n > 1 {\displaystyle n>1} , there always exists at least one prime number such that n < p < 2 n . {\displaystyle n | Infinitude of prime numbers |
c_7ou87izvt68u | In number theory, Bertrand's postulate is a theorem stating that for any integer n > 3 {\displaystyle n>3} , there always exists at least one prime number p {\displaystyle p} with n < p < 2 n − 2. {\displaystyle n 1 {\displaystyle n>1} , there is always at least one prime p {\displaystyle p} such that n < p < 2 n . {\d... | Bertrand's postulate |
c_3ri5igynnbj6 | In number theory, Bonse's inequality, named after H. Bonse, relates the size of a primorial to the smallest prime that does not appear in its prime factorization. It states that if p1, ..., pn, pn+1 are the smallest n + 1 prime numbers and n ≥ 4, then p n # = p 1 ⋯ p n > p n + 1 2 . {\displaystyle p_{n}\#=p_{1}\cdots p... | Bonse's inequality |
c_zwsvrepn5dxx | In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (pn)2 and (pn+1)2, where pn is the nth prime number, for every n ≥ 2. The conjecture is named after Henri Brocard. It is widely believed that this conjecture is true. | Brocard's conjecture |
c_miko3evlrbu9 | However, it remains unproven as of 2022. The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... OEIS: A050216. Legendre's conjecture that there is a prime between consecutive integer squares directly implies that there are at least two primes between prime squares for pn ≥ 3 since pn+1 − pn ≥ 2. | Brocard's conjecture |
c_pwya40o5dcb5 | In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by B2 (sequence A065421 in the OEIS). Brun's theorem was proved by Viggo Brun in 1919, and it has historical importa... | Brun's constant |
c_7y81ts9nk92t | In number theory, Büchi's problem, also known as the n squares' problem, is an open problem named after the Swiss mathematician Julius Richard Büchi. It asks whether there is a positive integer M such that every sequence of M or more integer squares, whose second difference is constant and equal to 2, is necessarily a ... | Büchi's problem |
c_7ai9d394hqil | In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind Un(P, Q) with relatively prime parameters P, Q and positive discriminant, an element Un with n ≠ 1, 2, 6 has at least one prime divisor that does not divid... | Carmichael's theorem |
c_crzgqn4eotqd | In number theory, Chebyshev's bias is the phenomenon that most of the time, there are more primes of the form 4k + 3 than of the form 4k + 1, up to the same limit. This phenomenon was first observed by Russian mathematician Pafnuty Chebyshev in 1853. | Chebyshev's bias |
c_io94zycpafne | In number theory, Chen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes). It is a weakened form of Goldbach's conjecture, which states that every even number is the sum of two primes. | Chen's theorem |
c_w13ulc6d7phd | In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be. It st... | Cramér conjecture |
c_s669qty9ims9 | In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers α {\displaystyle \alpha } and N {\displaystyle N} , with 1 ≤ N {\displaystyle 1\leq N} , there exist integers p {\displaystyle p} and q {\displaystyle q} such that 1 ≤ q ≤ ... | Dirichlet's theorem on diophantine approximation |
c_h2p29q4xil1f | This is a fundamental result in Diophantine approximation, showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality 0 < | α − p q | < 1 q 2 {\displaystyle 0<\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2}}}} is sa... | Dirichlet's theorem on diophantine approximation |
c_oaakboy1psh4 | In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d. Th... | Dirichlet's theorem on arithmetic progressions |
c_040iazlx08zi | In number theory, Dixon's factorization method (also Dixon's random squares method or Dixon's algorithm) is a general-purpose integer factorization algorithm; it is the prototypical factor base method. Unlike for other factor base methods, its run-time bound comes with a rigorous proof that does not rely on conjectures... | Dixon's factorization method |
c_u07mgic001u6 | In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a be an integer coprime to p. Then a p − 1 2 ≡ { 1 ( mod p ) if there is an integer x such that x 2 ≡ a ( mod p ) , − 1 ( mod p ) if there is no such integer. {... | Euler's criterion |
c_nn90okp69lzo | }}\end{cases}}} Euler's criterion can be concisely reformulated using the Legendre symbol: ( a p ) ≡ a p − 1 2 ( mod p ) . {\displaystyle \left({\frac {a}{p}}\right)\equiv a^{\tfrac {p-1}{2}}{\pmod {p}}.} The criterion first appeared in a 1748 paper by Leonhard Euler. | Euler's criterion |
c_1yw398z7tjbk | In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers, and φ ( n ) {\displaystyle \varphi (n)} is Euler's totient function, then a raised to the power φ ( n ) {\displaystyle \varphi (n)} is congruent to 1 modulo n; tha... | Euler's theorem |
c_qldmug5gk1u4 | The theorem is further generalized by Carmichael's theorem. The theorem may be used to easily reduce large powers modulo n {\displaystyle n} . For example, consider finding the ones place decimal digit of 7 222 {\displaystyle 7^{222}} , i.e. 7 222 ( mod 10 ) {\displaystyle 7^{222}{\pmod {10}}} . | Euler's theorem |
c_2bs978i2w61h | The integers 7 and 10 are coprime, and φ ( 10 ) = 4 {\displaystyle \varphi (10)=4} . So Euler's theorem yields 7 4 ≡ 1 ( mod 10 ) {\displaystyle 7^{4}\equiv 1{\pmod {10}}} , and we get 7 222 ≡ 7 4 × 55 + 2 ≡ ( 7 4 ) 55 × 7 2 ≡ 1 55 × 7 2 ≡ 49 ≡ 9 ( mod 10 ) {\displaystyle 7^{222}\equiv 7^{4\times 55+2}\equiv (7^{4})^{5... | Euler's theorem |
c_oiqnu7w8m07b | In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as φ ( n ) {\displaystyle \varphi (n)} or ϕ ( n ) {\displaystyle \phi (n)} , and may also be called Euler's phi function. In other words, it is the num... | Euler totient |
c_xzb8j68yuee6 | Therefore, φ(9) = 6. As another example, φ(1) = 1 since for n = 1 the only integer in the range from 1 to n is 1 itself, and gcd(1, 1) = 1. | Euler totient |
c_bh9dor2xh93s | Euler's totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ(mn) = φ(m)φ(n). This function gives the order of the multiplicative group of integers modulo n (the group of units of the ring Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } ). It is also used for... | Euler totient |
c_fp647j78z575 | In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} can satisfy the equation a n + b n = c n {\displaystyle a^{n}+b^{n}=c^{n}} for any integer value of n {\dis... | Mathematical conjecture |
c_qkeem8d11kbs | In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solu... | Fermat’s Last Theorem |
c_gdiqvijdvdp7 | Consequently the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles and formally published in 1995. | Fermat’s Last Theorem |
c_pzjn0a7os0zk | It was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016. It also proved much of the Taniyama–Shimura conjecture, subsequently known as the modularity theorem, and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. T... | Fermat’s Last Theorem |
c_h3j9wo3rexzz | In number theory, Glaisher's theorem is an identity useful to the study of integer partitions. Proved in 1883 by James Whitbread Lee Glaisher, it states that the number of partitions of an integer n {\displaystyle n} into parts not divisible by d {\displaystyle d} is equal to the number of partitions in which no part i... | Glaisher's theorem |
c_fvh5x6rpdowj | In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum. )This conjecture is called "weak" ... | Goldbach's weak conjecture |
c_9tuix8ggz8q0 | For if every even number greater than 4 is the sum of two odd primes, adding 3 to each even number greater than 4 will produce the odd numbers greater than 7 (and 7 itself is equal to 2+2+3). In 2013, Harald Helfgott released a proof of Goldbach's weak conjecture. As of 2018, the proof is widely accepted in the mathema... | Goldbach's weak conjecture |
c_jje66qov18do | The proof was accepted for publication in the Annals of Mathematics Studies series in 2015, and has been undergoing further review and revision since; fully-refereed chapters in close to final form are being made public in the process.Some state the conjecture as Every odd number greater than 7 can be expressed as the ... | Goldbach's weak conjecture |
c_ms3z05s8q6wp | In number theory, Grimm's conjecture (named after Carl Albert Grimm, 1 April 1926 – 2 January 2018) states that to each element of a set of consecutive composite numbers one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly, 76(1969) 1126-1128. | Grimm's conjecture |
c_7yt1j2kru547 | In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the ... | Hilbert's irreducibility theorem |
c_r9dlr8wly3pe | In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation. The theorem states that for every irrational number ξ there are infinitely many relatively prime integers m, n such that The condition that ξ is irrational cannot be omitted. Moreover the constant 5 {\displayst... | Hurwitz's theorem (number theory) |
c_55b7vdtg93zx | In number theory, Issai Schur showed in 1912 that for every nonconstant polynomial p(x) with integer coefficients, if S is the set of all nonzero values { p ( n ) ≠ 0: n ∈ N } {\displaystyle {\begin{Bmatrix}p(n)\neq 0:n\in \mathbb {N} \end{Bmatrix}}} , then the set of primes that divide some member of S is infinite. | Schur's theorem |
c_tayl03lfcd65 | In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa (1959) (岩澤 健吉), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generaliza... | Iwasawa theory |
c_pgycrdg11eo4 | In number theory, Kaprekar's routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. Each iteration starts with a number, sorts the digits into descending and ascending order, and calculates the difference between the two new numbers. As an example, starting with the number 8991... | Kaprekar's routine |
c_jype0ne2wnjv | In number theory, Lagrange's theorem is a statement named after Joseph-Louis Lagrange about how frequently a polynomial over the integers may evaluate to a multiple of a fixed prime. More precisely, it states that if p is a prime number, x ∈ Z / p Z {\displaystyle x\in \mathbb {Z} /p\mathbb {Z} } , and f ( x ) ∈ Z {\d... | Lagrange's theorem (number theory) |
c_d8hpps2dz4dc | In number theory, Lemoine's conjecture, named after Émile Lemoine, also known as Levy's conjecture, after Hyman Levy, states that all odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime. | Lemoine's conjecture |
c_8kc328qf0nj7 | In number theory, Lochs's theorem concerns the rate of convergence of the continued fraction expansion of a typical real number. A proof of the theorem was published in 1964 by Gustav Lochs.The theorem states that for almost all real numbers in the interval (0,1), the number of terms m of the number's continued fractio... | Lochs' theorem |
c_3azug4obkaqi | A prominent example of a number not exhibiting this behavior is the golden ratio—sometimes known as the "most irrational" number—whose continued fraction terms are all ones, the smallest possible in canonical form. On average it requires approximately 2.39 continued fraction terms per decimal digit. == References == | Lochs' theorem |
c_z4xwo4wo9pd4 | In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient ( m n ) {\displaystyle {\tbinom {m}{n}}} by a prime number p in terms of the base p expansions of the integers m and n. Lucas's theorem first appeared in 1878 in papers by Édouard Lucas. | Lucas' theorem |
c_6mqqh91n6gx5 | In number theory, Maier's theorem (Maier 1985) is a theorem about the numbers of primes in short intervals for which Cramér's probabilistic model of primes gives a wrong answer. The theorem states that if π is the prime-counting function and λ is greater than 1 then π ( x + ( log x ) λ ) − π ( x ) ( log x ) λ − 1 {... | Maier's theorem |
c_99qduqzxm974 | In number theory, Mazur's control theorem, introduced by Mazur (1972), describes the behavior in Zp extensions of the Selmer group of an abelian variety over a number field. | Mazur's control theorem |
c_c7sybvbwtcxy | In number theory, Meyer's theorem on quadratic forms states that an indefinite quadratic form Q in five or more variables over the field of rational numbers nontrivially represents zero. In other words, if the equation Q(x) = 0has a non-zero real solution, then it has a non-zero rational solution (the converse is obvio... | Meyer's theorem |
c_u1qms7e7aru2 | Meyer's theorem is usually deduced from the Hasse–Minkowski theorem (which was proved later) and the following statement: A rational quadratic form in five or more variables represents zero over the field Qp of the p-adic numbers for all p.Meyer's theorem is best possible with respect to the number of variables: there ... | Meyer's theorem |
c_0jvwjft8n7rp | In number theory, Mills' constant is defined as the smallest positive real number A such that the floor function of the double exponential function ⌊ A 3 n ⌋ {\displaystyle \lfloor A^{3^{n}}\rfloor } is a prime number for all positive natural numbers n. This constant is named after William Harold Mills who proved in 19... | Mills' constant |
c_mhh7s9fuarqz | In number theory, Moessner's theorem or Moessner's magic is related to an arithmetical algorithm to produce an infinite sequence of the exponents of positive integers 1 n , 2 n , 3 n , 4 n , ⋯ , {\displaystyle 1^{n},2^{n},3^{n},4^{n},\cdots ~,} with n ≥ 1 , {\displaystyle n\geq 1~,} by recursively manipulating the sequ... | Moessner's theorem |
c_wldy83ghh1ud | In number theory, Niven's constant, named after Ivan Niven, is the largest exponent appearing in the prime factorization of any natural number n "on average". More precisely, if we define H(1) = 1 and H(n) = the largest exponent appearing in the unique prime factorization of a natural number n > 1, then Niven's constan... | Niven's constant |
c_o3sgn8jrz9s3 | In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q {\displaystyle \mathbb {Q} } is equivalent to either the usual real absolute value or a p-adic absolute value. | Ostrowski's theorem |
c_smdssx14lh2h | In number theory, Poisson summation can also be used to derive a variety of functional equations including the functional equation for the Riemann zeta function.One important such use of Poisson summation concerns theta functions: periodic summations of Gaussians . Put q = e i π τ {\displaystyle q=e^{i\pi \tau }} , for... | Poisson summation formula |
c_9uskyc99zdr6 | In number theory, Proth's theorem is a primality test for Proth numbers. It states that if p is a Proth number, of the form k2n + 1 with k odd and k < 2n, and if there exists an integer a for which a p − 1 2 ≡ − 1 ( mod p ) , {\displaystyle a^{\frac {p-1}{2}}\equiv -1{\pmod {p}},} then p is prime. In this case p is cal... | Proth's theorem |
c_bp7rik55phei | In practice, however, a quadratic nonresidue of p is found via a modified Euclid's algorithm and taken as the value of a, since if a is a quadratic nonresidue modulo p then the converse is also true, and the test is conclusive. For such an a the Legendre symbol is ( a p ) = − 1. {\displaystyle \left({\frac {a}{p}}\righ... | Proth's theorem |
c_0yodbdvsfxau | Thus, in contrast to many Monte Carlo primality tests (randomized algorithms that can return a false positive), the primality testing algorithm based on Proth's theorem is a Las Vegas algorithm, always returning the correct answer but with a running time that varies randomly. Note that if a is chosen to be a quadratic ... | Proth's theorem |
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